Integer Solutions: 5x-3 < -3, -1 ≤ X ≤ 6

by Mei Lin 41 views

Hey guys! Today, let's dive into a fun math problem where we're going to find all the integer values of x that fit a certain inequality within a given range. It's like a little puzzle, and we'll solve it together step by step. So, grab your thinking caps, and let's get started!

Understanding the Problem

Before we jump into solving, let's make sure we understand what the problem is asking. We have an inequality: 5x - 3 < -3. This means we're looking for values of x that, when plugged into this expression, make the left side less than the right side. Think of it like a balancing scale, where we want the side with 5x - 3 to be lighter than the side with -3.

But there's a catch! We're not just looking for any value of x. We have a specific interval: -1 ≤ x ≤ 6. This tells us that x must be an integer (a whole number, like -2, -1, 0, 1, 2, etc.) and it has to fall between -1 and 6, inclusive. That means -1 and 6 are also potential solutions. So, we're essentially searching for integer values within this range that make our inequality true. This is a really crucial concept in mathematics because it combines the ideas of inequalities with the discrete nature of integers. When we're dealing with inequalities, we're often talking about a range of possible values, rather than just a single solution. And when we restrict ourselves to integers, we're looking at specific, countable values within that range. This kind of problem pops up all the time in various fields, from computer science (where you might be looking for the number of loop iterations) to economics (where you might be modeling quantities that can only be whole numbers).

So, to recap, we're on a mission to find the integer values of x that live between -1 and 6, and also make the expression 5x - 3 less than -3. Think of it like finding the hidden treasure within a specific area – a treasure that only certain keys (our integer values of x) can unlock. This constraint makes the problem more manageable because we're not dealing with an infinite number of possibilities. Instead, we have a finite set of integers to test, which makes our task much more concrete and achievable. Understanding this is key to tackling similar problems in the future, where you'll often encounter inequalities and constraints that limit the possible solutions to a specific set. So, now that we've got a clear picture of the problem, let's roll up our sleeves and start solving it!

Solving the Inequality

Okay, let's get down to business and solve this inequality! Our goal is to isolate x on one side of the inequality sign. We want to get x all by itself, so we can see what values make the inequality true. Remember, solving an inequality is very similar to solving an equation, but there's one important difference we'll need to keep in mind.

First, let's rewrite our inequality: 5x - 3 < -3. The first step is to get rid of that -3 on the left side. We can do this by adding 3 to both sides of the inequality. This is like adding the same weight to both sides of our balancing scale – it keeps things balanced. So, we get:

5x - 3 + 3 < -3 + 3

This simplifies to:

5x < 0

Great! We're one step closer to isolating x. Now, we have 5x on the left side, and we want just x. To do this, we need to divide both sides of the inequality by 5. And here's that important rule to remember: when you multiply or divide both sides of an inequality by a negative number, you have to flip the inequality sign. But in this case, we're dividing by a positive number (5), so we don't need to worry about flipping the sign. Phew!

So, let's divide both sides by 5:

5x / 5 < 0 / 5

This gives us:

x < 0

Awesome! We've solved the inequality. We now know that any value of x that is less than 0 will satisfy the inequality 5x - 3 < -3. This is a crucial piece of information, but we're not quite done yet. Remember, we have that interval to consider. We're not just looking for any x less than 0; we're looking for integer values of x that are less than 0 and fall within the range -1 ≤ x ≤ 6. Solving the inequality is like finding the general direction of the treasure, but the interval is like the map that tells us exactly where to dig. Without both pieces of information, we might be searching in the wrong place. This is why it's so important to pay attention to all the conditions and constraints given in a problem. They're not just there to make things complicated; they're essential clues that help us narrow down the possibilities and find the correct solution. So, now that we know x must be less than 0, let's take a look at our interval and see which integers fit the bill!

Identifying Integer Values within the Interval

Alright, we've cracked the inequality code and found that x must be less than 0. But remember, we're on a quest for integer values within the interval -1 ≤ x ≤ 6. This interval is like our playground – we can only pick numbers from within these boundaries. So, let's list out all the integers in this interval. It's a finite list, which makes our job much easier!

The integers in the interval -1 ≤ x ≤ 6 are: -1, 0, 1, 2, 3, 4, 5, 6. Notice that we include both -1 and 6 because the inequality signs include the “equal to” part (≤). If it were just “<”, we wouldn't include those endpoints.

Now, we have two pieces of the puzzle: x must be less than 0, and x must be one of these integers. It's time to put these pieces together and see which integers fit both conditions. Think of it like a Venn diagram – we have one circle representing numbers less than 0, and another circle representing the integers in our interval. The solution is where the circles overlap.

Let's go through our list of integers and see which ones are less than 0:

  • -1: Is -1 less than 0? Yes! So, -1 is a potential solution.
  • 0: Is 0 less than 0? No. 0 is equal to 0, but not less than.
  • 1: Is 1 less than 0? No.
  • 2: Is 2 less than 0? No.
  • 3: Is 3 less than 0? No.
  • 4: Is 4 less than 0? No.
  • 5: Is 5 less than 0? No.
  • 6: Is 6 less than 0? No.

It looks like only one integer from our list satisfies both conditions. That's the beauty of combining inequalities and intervals – it narrows down the possibilities and gives us a very specific solution. This process of identifying integers within a given range that satisfy a particular condition is a fundamental skill in many areas of mathematics. It's used in number theory, discrete mathematics, and even in calculus when dealing with sequences and series. So, understanding how to do this is a valuable tool in your mathematical toolkit. Now, let's state our final answer!

Stating the Solution

Drumroll, please! We've solved the inequality, we've identified the integers within the interval, and now it's time to state our solution loud and clear. We found that only one integer value of x satisfies both the inequality 5x - 3 < -3 and the interval -1 ≤ x ≤ 6.

That integer is -1. This is our final answer! We can write it as: x = -1.

To be absolutely sure, we can always double-check our answer. Let's plug -1 back into the original inequality and see if it works:

5(-1) - 3 < -3

-5 - 3 < -3

-8 < -3

This is true! -8 is indeed less than -3. So, we've confirmed that our solution is correct. High five!

This process of verifying your solution is a crucial step in problem-solving. It's like proofreading your work before submitting it. It helps you catch any mistakes and ensures that your answer is accurate. In mathematics, verification is often just as important as finding the solution itself. It builds confidence in your answer and demonstrates a thorough understanding of the problem.

So, to recap, we started with an inequality and an interval, we solved the inequality to find the general condition for x, we identified the integers within the interval, and then we combined these two pieces of information to find the specific integer solution. And we verified our answer to make sure it was correct. That's a lot of math in one problem! But by breaking it down into smaller steps, we were able to tackle it successfully. This approach – breaking down complex problems into smaller, manageable steps – is a valuable strategy not just in mathematics, but in many areas of life. So, remember this process, and you'll be well-equipped to solve all sorts of challenges. Great job, guys! We conquered this math problem together. Now, let's move on to the next adventure!

Conclusion

So, there you have it! We successfully navigated the world of inequalities and intervals to find the integer solution. Remember, the key is to break down the problem into smaller, manageable steps. First, solve the inequality to get a general idea of the possible values of x. Then, consider the given interval to narrow down the possibilities to a specific set of integers. Finally, combine these two pieces of information to identify the integer values that satisfy both conditions. And don't forget to verify your solution to make sure it's correct!

This type of problem is a great example of how different mathematical concepts come together. It's not just about knowing how to solve an inequality; it's also about understanding integers, intervals, and how to combine these ideas to solve a more complex problem. These are skills that will serve you well in future math courses and in various real-world applications. Whether you're calculating the optimal number of products to manufacture, determining the range of acceptable values in a scientific experiment, or even just figuring out how many slices of pizza to order, the ability to work with inequalities and intervals is a valuable asset. So, keep practicing, keep exploring, and keep having fun with math! You guys are doing awesome!